In silicon integrated circuit devices, to introduce impurities into a silicon substrate, ion implantation is generally performed. In process construction of such silicon integrated circuit devices, to obtain necessary device structure, it is necessary to determine ion implantation conditions. To that end, simulation is generally performed to determine the ion implantation conditions.
In a cutting-edge MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor), research is being conducted to activate while maintaining the ion implantation distribution using the co implantation technique or Flash lamp anneal technique. Namely, the trend is that complicated diffusion phenomena will no longer have to be considered. Until recently, a diffusion distribution far shifted from the ion implantation distribution has been a main factor to determine the device characteristics. However, when such diffusion can be ignored, it becomes more and more important to precisely predict (estimate) two-dimensional ion implantation distribution.
On the other hand, inverse modeling is also used which predicts an impurity distribution reproducing obtained electrical characteristics separately from process conditions (i.e., without the process conditions). In the inverse modeling, though it has little relationship with the process conditions, it becomes possible to obtain knowledge of impurity distribution separately from an unestablished process model. In this case, the Gauss distribution or the complementary error function distribution is used. By optimizing the parameter of the function, the electric characteristics are reproduced. After that, it becomes possible to predict how to change the distribution to realize the improvement of the device characteristics.
When the tilt angle is small, a general model using such a Gauss distribution is expressed in the following formulas (1) to (4) while assuming that the incidence is normal and that the coordinates (x,y) of the ion implantation system are the same as the coordinates (t,s) of an actual device.
(i) Substrate
In a substrate, it is assumed that the concentration is constant as described in the following formula (1).N1(x,y)=NsubN0(x,y)=Nsub  (1)(ii) Channel Ion Implantation Distribution
Since ions are ejected to the front face while no gate electrode is formed, the distribution is expressed in the following formula (2).
                                          N            1                    ⁡                      (                          x              ,              y                        )                          =                              Φ                                                            2                  ⁢                  π                                            ⁢              Δ              ⁢                                                          ⁢                              R                p                                              ⁢                      exp            ⁡                          [                              -                                                                            (                                              y                        -                                                  R                          p                                                                    )                                        2                                                        2                    ⁢                    Δ                    ⁢                                                                                  ⁢                                          R                      p                      2                                                                                  ]                                                          (        2        )            (iii) Extension Region Ion Implantation Distribution
Assuming that the gate electrode having the length of LG is formed, the distribution is expressed in the following formula (3).
                                          N            2                    ⁡                      (                          x              ,              y                        )                          =                              [                          1              -                                                                    erf                    (                                                                                                                        L                            G                                                    2                                                -                        x                                                                                              2                                                ⁢                        Δ                        ⁢                                                                                                  ⁢                                                  R                                                      p                            ⁢                                                                                                                  ⁢                            t                                                                                                                )                                    +                                      erf                    (                                                                                                                        L                            G                                                    2                                                +                        x                                                                                              2                                                ⁢                        Δ                        ⁢                                                                                                  ⁢                                                  R                                                      p                            ⁢                                                                                                                  ⁢                            t                                                                                                                )                                                  2                                      ]                    ⁢                      Φ                                                            2                  ⁢                  π                                            ⁢              Δ              ⁢                                                          ⁢                              R                p                                              ⁢                      exp            ⁡                          [                              -                                                                            (                                              y                        -                                                  R                          p                                                                    )                                        2                                                        2                    ⁢                    Δ                    ⁢                                                                                  ⁢                                          R                      p                      2                                                                                  ]                                                          (        3        )            (iv) Source/Drain Region Ion Implantation Distribution
Assuming that side walls having the thickness of Lside are formed on both sides of the gate electrode, the distribution is expressed in the following formula (4).
                                          N            3                    ⁡                      (                          x              ,              y                        )                          =                              [                          1              -                                                                    erf                    (                                                                                                                                                      L                              G                                                        +                                                          2                              ⁢                                                              L                                side                                                                                                              2                                                ⁢                        x                                                                                              2                                                ⁢                        Δ                        ⁢                                                                                                  ⁢                                                  R                                                      p                            ⁢                                                                                                                  ⁢                            t                                                                                                                )                                    +                                      erf                    (                                                                                                                                                      L                              G                                                        +                                                          2                              ⁢                                                              L                                side                                                                                                              2                                                ⁢                                                                                                  +                        x                                                                                              2                                                ⁢                        Δ                        ⁢                                                                                                  ⁢                                                  R                                                      p                            ⁢                                                                                                                  ⁢                            t                                                                                                                )                                                  2                                      ]                    ⁢                      Φ                                                            2                  ⁢                  π                                            ⁢              Δ              ⁢                                                          ⁢                              R                p                                              ⁢                      exp            ⁡                          [                              -                                                                            (                                              y                        -                                                  R                          p                                                                    )                                        2                                                        2                    ⁢                    Δ                    ⁢                                                                                  ⁢                                          R                      p                      2                                                                                  ]                                                          (        4        )            
In formulas, the symbols “δ”, “Rp”, “ΔRp”, and “ΔRpt” denote a dose amount, projection of a range, straggling of the projection of the range, and straggling of the projection of the range in lateral direction, respectively.
Then, signs are added depending on the types of the distributions and the summing operation is performed. Depending on the processes, there may be a case where the order of ion implantations changes and the timing of forming the side wall also changes. However, by appropriately changing the parameters in the formulas, it may be possible to correspond to those changes. By considering the types of the distributions and performing the summing operation, the impurity concentration distribution Nnet may be obtained as given in the following formula (5) as the final distribution.
                              N          net                =                              ∑            i                    ⁢                                    sgn              i                        ⁢                          N              i                                                          (        5        )            
Where, the symbol “sgni” denotes a function expressing the type and is given as in the following formula (6).
                              sgn          i                =                  {                                                    1                                                                                  N                    i                                    ⁢                                      :                                    ⁢                                                                          ⁢                  donor                                                                                                      -                  1                                                                                                  N                    i                                    ⁢                                      :                                    ⁢                                                                          ⁢                  acceptor                                                                                        (        6        )            
When it becomes possible to analytically express the entire ion implantation distribution for the MOS structure like a Gauss distribution, it may become possible to easily perform the device simulation. Further, such a distribution may also be applied to the inverse modeling. To that end, it is preferable to simply and analytically express the channel ion implantation distribution, the extension region ion implantation distribution, the source/drain region ion implantation distribution, and a pocket ion implantation distribution. Especially, it may be important to associate the pocket ion implantation distribution with the actual process. This is because the pocket ion implantation distribution may directly influence the short-channel resistance, the on-state current, and the off-sate current of the device and a main part of the device characteristics.
As described above, when a tilt angle is small, the two-dimensional ion implantation distribution may be expressed by using a simple analytical model when the range projection straggling in lateral direction (ΔRpt) is obtained (see, for example, Patent Document 1, Non-Patent Documents 1 to 3).    Patent Document 1: Japanese Laid-open Patent Publication No. 2003-163173    Non-Patent Document 1: J. C. Hu, A. Chaterjee, M. Mehrotra, J. Xu, W.-T. Shiau, and M. Rodder, “Sub-0.1 mm CMOS source/drain extension spacer formed using nitrogen implantation prior to thick gate re-oxidation,” Sysmposium on VLSI Tech., pp. 488-189, 2000.    Non-Patent Document 2: Y. Momiyama, K. Okabe, H. Nakao, M. Kojima, M. Kase, and T. Sugii, “Extension engineering using carbon co-implantation technology for low power CMOS design with phousphorus- and Boron-extension,” Ext. abs. The 7th International Workshop on Junction Technology, pp. 63-64, 2007.    Non-Patent Document 3: P. A. Stolk, D. J. Eaglesham, H.-J. 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However, until recently, the actual two-dimensional ion implantation distribution has been evaluated only numerical-analytically by using a two-dimensional process simulator. Because of this feature, there may be a problem that, when the tilt angle is large, the ion implantation distribution in the MOS structure is not analytically evaluated. Further, the actual distribution is different from the Gauss distribution employed in the inverse modeling. Because of this feature, even when qualitative guidelines are given, it may be difficult to directly associate the actual distribution with the actual process conditions. Especially, there is no appropriate analytical model for the pocket ion implantation distribution in which ions are implanted at a large tilt angle. Accordingly, it may be difficult to get feedback from the distribution used in the inverse modeling to the actual process conditions.
Further, when assuming that the distribution along the axis of ion implantation is the Gauss distribution, the analytical model as described above may be obtained. However, such an analytical model may not be obtained when the distribution is the Pearson distribution generally used or the tail functions proposed by the inventor of the present invention (see, for example, Non-Patent Document 4).
Because of this feature, there may be problems such that strong expertise is needed to use the process simulator or the device simulator and that a vast amount of time is necessary for the simulations.
However, for developers involved in various devices and processes, sophisticated functions may not be needed, and there is a demand for providing a simulator as a tool providing for developer's experiences and intuitive settings. The simulator in this case is needed to be easily operated and provide fast operations. Further, the functions used in the simulator are required to be accurately operated even when the numbers of the functions are limited.